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I. Bentley and S. FrauendorfDepartment of Physics
University of Notre Dame, USA
Calculation of the Wigner Term in the Binding Energies by Diagonalization of
the Isovector Pairing Hamiltonian
Eground bvolume A bsurface A2 / 3
E shell (N,Z) bCoulomb
Z 2
A1/ 3 cCoulombZZ
A
1/ 3
bsymmetry
| Tz | (| Tz |X)
A
Tz N Z
2
bsymmetry 92MeV , A 100
linear term : ~ 1MeV| Tz |
Important for p/rp process near the N=Z line
Quantify X
Subtract the Coulomb energy
ECoulomb bCoulomb
Z 2
A1/ 3 cCoulombZ
Z
A
1/ 3
ECoulomb
Z A
2bCoulomb
Z
A1/ 3 cCoulomb
1
2
1/ 3
E
Z mirror nuclei
Estrong Eground ECoulomb
A=44
A=56
A=68
2
)1()1(
||2/
||
|)(|
)|(|||2
:chain isobarican Along
zstrongzstrong
z
shellz
z
strong
zshell
zzsymmetrystrong
TETE
T
EXTc
T
E
constTEA
XTTbE
“Experimental” Wigner X
Substantial scatter caused by shell effectsMean value ~1 for A<70Mean value ~4 for 80<A<90
Contains shell effects! Separation is problematic.
Eground bvolumeA bsurfaceA2 / 3
E shell (N,Z) bCoulomb
Z 2
A1/ 3 cCoulombZZ
A
1/ 3
bsymmetry(A)| Tz | (| Tz |X(A))
A
Tz N Z
2P. Moeller et al. Atomic Data and
Nuclear Data Tables 59, 185 (1995):
Phenomenological expression for symmertry energy,
which corresponds to X 1
Phenomenological treatment: Micro-Macro
Density functionals
Skyrme–Hartree–Fock–Bogoliubov mass formula by N. Chamel, S. Goriely, J.M. Pearson, Nuclear Physics A 812 (2008) 72–98:
Skyrme HFB give parameter dependent values of X, substantially smaller than 1, sensitive to effective mass (Satula, Wyss, Rep. Prog. Phys. 68, 131 (05)
Unsatisfactory!
Relativistic Mean Field gives X approximately 1(Ban et al., Phys. Lett. B 633, 231 (06)
What is the origin of X?
Isovector Proton-Neutron Pairing.
Strength is fixed by isospin invariance of stronginteraction. It gives X approximately 1 by symmetry.(Frauendorf, Sheikh, Nucl. Phys. A 645, 509, (99)
1) Fixing the isovector pairing strength to the standard value for pp, nn pairing, obtained from even-odd mass differences, we quantitatively reproduce the experimental X.
2) Possibilities for implementation into density functional approaches (ongoing)
There is a well founded mechanism, which has to be there:
Isovector Pairing Hamiltonian
Generate all configurations by lifting pp, nn, pn pairs and diagonalize.6 or 7 levels around the Fermi level -> dimension ~ 10000 Few cases with 8 levels -> no significant change if G is scaled.
H (k, )k, ˆ N k, G ˆ P k,
ˆ P k,
ˆ P k, 1 ˆ p k
ˆ p k ˆ P k,1
ˆ n k ˆ n
k
ˆ P k,0
1
2ˆ n k ˆ p
k ˆ p k
ˆ n k
[H,r T ]0 : Isospin is conserved.
Solve the pairing problem by diagonalization:-Isospin is good-No problems with instabilities of the pair field
Why X=1? Strong pairing limit Spontaneous breaking of isorotational symmetry
hmf (k, )k, ˆ N k
ˆ P k, ˆ P k,
k,
quasiparticles mixed from proton and neutron particles and holes
ˆ P k,
k
Pair field is a vector, which spontaneously breaks the isorotational symmetry.
Since [Tx ,Ty ] iTz (SU2) strong breaking generates an isorotational band
E(T)T(T 1)
"strong" means >> level distance
Frauendorf SG, Sheikh JACranked shell model and isospin symmetry near N=Z NUCLEAR PHYSICS A 645, 509 (1999)
Isorotations (strong symmetry breaking)Bayman, Bes, Broglia, PRL 23 (1969) 1299 ( 2 particle transfer)Frauendorf, Sheikh, NPA 645, 509 (1999) Frauendorf, Sheikh, Physica Scripta T88, 162 (2000)
intrinsic state : | Described by common
quasi proton and quasi neutron excitations without an pn - pair field
isorotational state : DTz
T
0( ,,0) |
isorotational bands : E(T,Tz) hmf Tz T(T 1)
2Spectra of deformed N Z nuclei organanize into
spatio - iso - rotational bands.
Afanasjev AV, Frauendorf S, PRC 71, 064318 (2005)Afanasjev AV, Frauendorf S, NPA 746, 575C (2004 )Kelsall NS, Svensson CE, Fischer S, et al. EURO. PHYS. J. A 20, 131 (2004)….
Wigner X with AutoTAC Deformations
• Not perfect, but promising.• Two problems :
44≤A≤58 too strong scatter74≤A≤88 Xc~1 Xe~4
• Why?• Calculated deformations • not good enough
smallmediumlarge
Rotational response
Optimize the deformation
• Nilsson calculated• Woods Saxon calculated•Folded Yukawa calculated•Experimental (BE2(2->0)•Experimental yrast energies
“adopted deformations”
• Isovector proton neutron pairing with the strength fixed by isospin conservation gives the correct X•Mean field treatment (HFB) is insufficient – violates isospin conservation•In devising approximations beyond mean field it is decisive to incorporate restoration of isospin
Isovector and isoscalar pairing
H (k, )k, ˆ N k, GV
ˆ P k, ˆ P k,
,k
GSˆ Q k ˆ Q k
k
ˆ P k, 1 ˆ p k
ˆ p k ˆ P k,1
ˆ n k ˆ n
k
ˆ P k,0 1
2ˆ n k ˆ p
k ˆ p k
ˆ n k
ˆ Q k
1
2ˆ n k ˆ p
k ˆ p k
ˆ n k
• Isoscalar pairing attenuates the staggering between the even-even and odd-odd N=Z nuclei: some indication from experiment•Small isoscalar pair correlation would only slightly increase the X values: within the tolerance range of the isovector scenario•What is GS/GV ?
Implementation into mean field approaches
•8 levels around the Fermi level is not enough-> dimensions explode->approximations.•Iso-cranking approximation•HFB + RPA•HFB + SCRPA•T-,N-,Z- projected HFB•BCS-truncation
Iso-cranking Frauendorf, Sheikh, NPA 645, 509 (1999)
For spatial rotations of well deformed nuclei do HFB with:
H 'H Jx, with constraint | Jx | >= I(I 1)
E | H | >I(I 1)
2
In analogy do HBF with:
H 'H Tz ˆ A ,
with constraints , | Tz |, >= T(T 1), , | ˆ A |, >= A
or equivalentely :
H 'H 1ˆ N 1
ˆ Z
with constraints , | ˆ N |, >= N , , | ˆ Z |, >= Z
T(T 1) T 1/2 and N Z
Problem: It works only for a sufficiently strong pair field.
HFB+Lipkin-Nogami may mend the problem.
HFB+QRPA
H (k, )k, ˆ N k, G ˆ P k,
ˆ P k,
2
r T
r T
ˆ P k, 1 ˆ p k
ˆ p k ˆ P k,1
ˆ n k ˆ n
k
ˆ P k,0
1
2ˆ n k ˆ p
k ˆ p k
ˆ n k
[H,r T ]0 : Isospin is conserved.
K. Neergard PLB 537, 287 (2002); PLB 572, 159 (2003); PRC 80, 044313 (2009)
T-, N-, Z-, projected HFB
| T,Tz,N,Z sinddT ,TT ()e iTy d
0
2
d0
2
e i(N ˆ N ) i(Z ˆ Z )
0
(uk vkPk ) | 0
k,1
vk
T,Tz,N,Z | H | T,Tz,N,Z T,Tz,N,Z | T,Tz,N,Z
0, uk2 uk
2 1
Simplified version: projected BCS
uk
vk
1
21
ek
ek 2 2
1/ 2
, 2 vk2 ( , )
Z, 1
N, 1k
E(1, 1) T,Tz,N,Z | H | T,Tz,N,Z T,Tz,N,Z | T,Tz,N,Z
minimum
Only nn pairing
In BCS state a certain configuration has the weight : w fk with fk uk if no pair
vk if pair
k
All configuarations with w taken.
BCS
BCS